It has been my good fortune in life to have been able to work with inspiring and generous people. When I was writing this address, I found myself thinking particularly of one, Geoffrey Matthews, to whom I still often turn when I need reminding of the funnier aspects of mathematics education. I have vivid memories of his elation, 15 years ago, when he was first approached to be President-Elect of the Mathematical Association. Being a stickler for punctuality, he went straight home and wrote his Presidential Address , more than one year in advance of its delivery. In contrast I started on this address just 10 days ago, and have had to amend it twice since, because of announcements of radical shifts in government policy. Even the title, chosen five months ago, may be out of date for reason which I will explain later.

Marilyn vos Savant is listed in the Guinnes Book of World Records Hall of Fame as the human being with the highest IQ; she makes herself useful by answering questions in her column Ask Marilyn, which appears, for example, in Parade Magazine, a publication inserted into the Sunday editions of several American newspapers.

1. An automorphic number is generally understood to be an integer the final digits of whose square (and therefore of any integral power) form the original number, i.e. a is automorphic if a2 ≡ a (mod 10n), where n is the number of digits of a. There are several references to these in books on recreational mathematics, some using also bases other than 10, but otherwise generally confined to definitions and examples. Investigating some elementary properties that I had not seen treated in print led me to submit an article to the Gazette, which was published in October 1989 (no. 465, p 212). This caught the attention of Dr R. J. Cook, of Frome, who kindly sent me the results of some experiments it had led him to make on applying the idea to indices > 2. These results were surprising and at first puzzling, but a lengthy subsequent correspondence has led to the emergence of a remarkably complicated and quite fascinating picture, which this article is an attempt to summarise.

Note 74.11 from Nick Lord in the March 1990 Gazette looks at the fascinating Simpson's paradox starting from a complex number situation; and in the same issue John Haigh, reviewing Cox and Snell's Analysis of Binary Data, gives a (fictitious) example; but the fascination of the paradox stems from the real occurrences.

Robert Harbin, in Waddingtons Family Card Games, gives an item called Prediction. You ask a friend to name two values of cards and to shuffle a deck. Then you bet that his two values will come up together as the deck is turned over. Harbin says he has worked out the odds as 32 to 19 in favour of success— “but I could be wrong. See what you make it.”

Mathematical magic, properly used, can play an important role in the mathematics classroom. Students need to be motivated to study mathematics (or any subject), and unusual and unexpected results, though sometimes frivolous, often provide more effective motivation than the utilitarian ideas of real world applications. Such an idea was discussed by Anderson in his article on the twenty-one card trick.

Perhaps the most extensive use of dimensional analysis in applied mathematics is in the application of the principle of similitude to obtain equivalence between different physical systems of unequal scales occurring in widely different branches of science and engineering. However, there is a widespread misconception that dimensional analysis can only yield the functional dependence of a physical quantity on various other parameters. In this article I shall therefore demonstrate some hidden strengths of dimensional analysis.

In French law, the 12 members of a jury are not required to return a unanimous verdict. It was decreed in 1831 that a vote of 8 or more would suffice. Four years later, it was proposed to reduce the requirement to 7 or more. There was a heated discussion in the Chamber of Deputies of this proposal which was strongly challenged by Arago, a mathematical physicist. Despite his ardent use of probability theory, the law was in fact changed.

In her Gazette article on “Roll-a-penny probabilities”, Ann Hirst sets out to find the probability that a player will lose after a certain number of rolls. She considers only the simplest case where the player starts with one penny, and solves the problem by setting up a recurrence relation and using induction. The problem of what happens if the player starts with more than one penny is left for the further consideration of the reader.

The aim of this note is to discuss the relative values of different ways of posing questions on the same material. I then try to decide which methods are most effective as assessment questions. The environment of examination room, home, availability of books and notes are, of course, crucial to the selection of a method. The time available for marking must also form a realistic constraint on the choice of question.